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IE 360 – STATISTICAL FORECASTING AND TIME SERIES

2020-2021 TERM PROJECT

INSTRUCTOR: MUSTAFA GÖKÇE BAYDOĞAN

PROJECT GROUP: GROUP 2

Disclamer: The data imported in this Project is credited to Trendyol.com ®

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TABLE OF CONTENTS

INTRODUCTION…………………………………………………………………………. 3

MODELS…………………………………………………………………………………… 3

OVERVIEW OF THE DATA…………………………………………………………….. 4

PRODUCTS AND ACCOMPANIED MODELS………………………………………… 5

CONCLUSION……………………………………………………………………………... 17

APPENDIX……………………………………………………………………………….....18

REFERENCES……………………………………………………………………………..28

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INTRODUCTION

Statistical forecasting refers to making predictions for future values based on past data.

It employs methods such as time series, cross-sectional or longitudinal data. For the purpose

of this project, time series are used to estimate the future sales of eight products on sale on

Trendyol.com. To analyze the data, it must be converted into time series. Time series analysis

is useful in a way to extract relevant data to the product being forecasted and other

characteristics, such as seasonality, trends, and autocorrelation. After embarking on with the

analysis, one is able to see the critical points of the data. After the analysis, several models

should be considered to fit the data based on multiple values. These values, such as mean

absolute percentage error, and mean absolute error, lead us to the best model fitting the data

of the product. Moving on with the model selection, the critical points then can be eliminated

if it’s considered to be affecting the future values. For example, if a holiday season has a

crucial effect on the sales, the effect should be removed from the model as it resonates on

daily regular sales. Finally, when the elimination is done, every factor is included and the

accompanying model is settled, the model can be used to forecast future sales. The means

through is R coding language for forecasting in this project.

MODELS

Several models can be used to fit a time series data. These models include moving

average models, exponential moving average models, naive model, autoregressive models,

ETS models, ARIMA models, single and multiple regression models, dynamic regression

models, and approaches to generate coherent forecasts such as bottom to top, top to bottom

and middle-out approaches.

Moving average models employs linearity to forecast a future value. It’s used for

univariate time series. The future predictions are based on solely the linearity of the past and

current data. As a type of a moving average model, exponential moving average model

utilizes weights on the past data. The most recent data is corresponded with a larger weight to

forecast the future values. Moving average models are a type of naive model. Naive models

don’t require any underlying assumptions other than the past data. Factors such as

autocorrelation, seasonality, trends, and special dates are not included in naive models.

Contrary to moving average models, Autoregressive models do not necessarily have to

be stationary. Autoregressive models are used to explain random occurences. The model

suggests that the forecast depends on the linearity of past data and a stochastic term. A

stochastic term is a random term and it’s impossible to predict the term fully correct.

Autoregressive models are a component of autoregressive–moving-average, and

autoregressive integrated moving average models. Autoregressive integrated moving average

model or ARIMA is crucial in our case. ARIMA model benefits the forecast by removing

non-stationarity using differencing once, or a several times depending on the data. ARIMA

models are generally denoted as ARIMA(p,d,q). The p in the parentheses meaning the number

of time lags, d meaning the number of times the data is differenced to remove non-

stationarity, and finally, q corresponding to the moving average term. ARIMA models can be

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estimated by using the Box-Jenkins Method. However, the R function, auto.arima(), can be

used to model the data automatically.

Regression models consist of including factors in the model that alters the future

predictions. For a single regression model, the most significant factor is selected after

performing an ANOVA test. After the selection, the model is set by R code lm(x ~ y, data).

With multiple regression, several factors can be included in the model after performing an

ANOVA test. With factors having high correlation, the less significant one should be removed

to maintain integrity in the model. Dynamic regression explains an additional random factor’s

effect on the initial prediction over time.

ETS models consist of error, trend, seasonality, as the abbreviation suggests. All ETS

models are based on non-stationary data while ARIMA models can be stationary. The model

focuses on seasonality and trends on time series univariate.

Lastly, coherent forecast approaches apply to multiple products, and their respective

families. Mentioned approaches only work with strictly hierarchical aggregation structures.

These approaches don’t work with groups. They involve first generating forecasts for the

series on hand, and then disaggregating these down the hierarchy for the top-down and up

the hierarchy for the bottom-up approaches until the last item. The middle-out approach

combines bottom-up and top-down approaches. Firstly, a middle layer is chosen and

forecasts are generated for all the series at this level. Then, from the layer, bottom-up and

top-down approaches are used for the forecast.

OVERVIEW OF THE DATA

The data from the eight products is imported to RStudio and converted to time

series. At first glance, the data from all the products seem completely random with partial

seasonality, and trends. After checked for characteristics, plots show that every single one

of them contain traces of seasonality and trend as well as a few outliers. Plots show no

sinusoidal functions, thus, no harmonic movement in the data. On a few occasions,

timeseries data shows that sales go up irregularly. These specific dates correspond to

holiday seasons. To check whether holiday seasons affect the data, we use regression

models. The holiday seasons we tested are as follows, Black Friday, Valentine’s Day,

Mother’s Day, Father’s Day, Ramadan, and New Year’s. After setting the regression model

for all products, using the summary() function in R, the only holiday that is significant is

seen to be Black Friday. Thus, for all product models, Black Friday season is excluded

from the data as it affects the regular sale predictions. Similar to holiday seasons, the other

data columns in the time series is checked if they had an excessive effect on the sales.

Dynamic regression is used to evaluate the claim with basket count, visit count, and add

count. However, since they are all interdependent, the dynamic regression model shows no

significance whatsoever between the columns.

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Figure 1: Daily Sales of Product 31515569 Over Time

PRODUCTS AND ACCOMPANIED MODELS

For each product, there’s a different model that fits better than the others. For each

product, simple linear regression, linear regression, arima model, arima model with linear

regression, STLF model (Seasonal and Trend decomposition using Loess Forecasting model),

and decomposition is tested to select the best model fit. Selection is solely dependent on the

MAPE value. A lower MAPE value indicates that the model fits better to the time series data,

since MAPE shows the mean absolute percentage error, it makes sense that a low MAPE

value is an indication of a good model fit. The results for each product are as follows,

Product 31515569

variable mse mad mape

1 predicted_slr Inf 7.236904e+246 8.424801e+243

2 predicted_lr Inf 7.236904e+246 8.424801e+243

3 predicted_arima 33687.60 1.392713e+02 3.013369e-01

4 predicted_lr_arima 11693.54 8.112254e+01 1.661093e-01

5 predicted_stlf 80690.35 2.621942e+02 7.060651e-01

6 predicted_decom 51161.78 2.039650e+02 5.663967e-01

Table 1: MSE, MAD, MAPE Values from Different models of Product 31515569

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From the table, the lowest MAPE value corresponds to ARIMA model with linear

regression. Thus, it’s the best fit for the product.

Figure 2: Forecast in Units Over Time for Product 31515569 with the Best Model Fit

Figure 3: Forecast Comparison with Other Models for Product 31515569

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Product 32939029

variable mse mad mape

1: predicted_slr 1.431854e+57 8.461250e+27 Inf

2: predicted_lr 1.431854e+57 8.461250e+27 Inf

3: predicted_arima 4.149693e+02 1.511438e+01 Inf

4: predicted_lr_arima 5.740437e+01 5.493045e+00 Inf

5: predicted_stlf 3.694727e+03 5.640661e+01 Inf

6: predicted_decom 6.640471e+03 7.739839e+01 Inf

Table 2: MSE, MAD, MAPE Values from Different models of Product 32939029

From the table, the lowest MAD value corresponds to ARIMA model. Thus, it’s the

best fit for the product. MAPE values are not used in comparison for this case as their value

doesn’t differ from each other.

Figure 4: Forecast in Units Over Time for Product 32939029 with the Best Model Fit

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Figure 5: Forecast Comparison with Other Models for Product 32903929

Product 3904356

variable mse mad mape

1: predicted_slr 7.073758e-01 0.841028664 Inf

2: predicted_lr 7.073758e-01 0.841028664 Inf

3: predicted_arima 3.548028e-06 0.001880499 Inf

4: predicted_lr_arima 4.760970e-04 0.020599287 Inf

5: predicted_stlf 2.814758e-04 0.011602629 Inf

6: predicted_decom 5.036980e-03 0.063144919 Inf

Table 3: MSE, MAD, MAPE Values from Different models of Product 3904356

From the table, the lowest MAD value corresponds to ARIMA model. Thus, it’s the

best fit for the product. MAPE values are not used in comparison for this case as their value

does not differ from each other.

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Figure 6: Forecast in Units Over Time for Product 3904356 with the Best Model Fit

Figure 7: Forecast Comparison with Other Models for Product 3904356

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Product 4066298

variable mse mad mape

1: predicted_slr Inf Inf Inf

2: predicted_lr Inf Inf Inf

3: predicted_arima 33109.294 118.98932 0.5806212

4: predicted_lr_arima 3058.023 48.40104 0.3343337

5: predicted_stlf 30419.651 109.15271 0.5026051

6: predicted_decom 37457.735 108.13554 0.4438671

Table 4: MSE, MAD, MAPE Values from Different models of Product 4066298

From the table, the lowest MAPE value corresponds to ARIMA model with Linear

Regression. Thus, it’s the best fit for the product.

Figure 8: Forecast in Units Over Time for Product 4066298 with the Best Model Fit

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Figure 9: Forecast Comparison with Other Models for Product 4066298

Product 5926527

variable mse mad mape

1: predicted_slr 3.893821e+09 1.406316e+04 Inf

2: predicted_lr 3.893821e+09 1.406316e+04 Inf

3: predicted_arima 1.626841e+00 9.510262e-01 Inf

4: predicted_lr_arima 2.600669e+00 9.967404e-01 Inf

5: predicted_stlf 2.239371e+00 1.130881e+00 Inf

6: predicted_decom 2.073538e+00 1.021640e+00 Inf

Table 5: MSE, MAD, MAPE Values from Different models of Product 5926527

From the table, the lowest MAD value corresponds to ARIMA model. Thus, it’s the

best fit for the product. MAPE values are not used in comparison for this case as their value

does not differ from each other.

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Figure 10: Forecast in Units Over Time for Product 5926527 with the Best Model Fit

Figure 11: Forecast Comparison with Other Models for Product 5926527

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Product 6676673

variable mse mad mape

1: predicted_slr Inf 3.536131e+206 4.830780e+203

2: predicted_lr Inf 3.536131e+206 4.830780e+203

3: predicted_arima 13961.322 8.915444e+01 2.886000e-01

4: predicted_lr_arima 7172.427 6.871835e+01 2.078257e-01

5: predicted_stlf 14976.947 8.304279e+01 2.165356e-01

6: predicted_decom 20886.263 1.265166e+02 4.457992e-01

Table 6: MSE, MAD, MAPE Values from Different models of Product 6676673

From the table, the lowest MAPE value corresponds to ARIMA model. Thus, it’s the

best fit for the product.

Figure 12: Forecast in Units Over Time for Product 6676673 with the Best Model Fit

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Figure 13: Forecast Comparison with Other Models for Product 6676673

Product 7061886

variable mse mad mape

1: predicted_slr 1.365059e+142 2.612526e+70 1.920975e+68

2: predicted_lr 1.365059e+142 2.612526e+70 1.920975e+68

3: predicted_arima 1.654208e+03 2.672492e+01 4.093273e-01

4: predicted_lr_arima 3.940499e+02 1.281498e+01 1.968616e-01

5: predicted_stlf 2.043986e+03 2.830530e+01 3.548158e-01

6: predicted_decom 1.517635e+03 2.647109e+01 4.178958e-01

Table 7: MSE, MAD, MAPE Values from Different models of Product 7061886

From the table, the lowest MAPE value corresponds to ARIMA model. Thus, it’s the

best fit for the product.

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Figure 14: Forecast in Units Over Time for Product 7061886 with the Best Model Fit

Figure 15: Forecast Comparison with Other Models for Product 7061886

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Product 85004

variable mse mad mape

1: predicted_slr 1.001791e+64 2.238154e+31 2.572594e+29

2: predicted_lr 1.001791e+64 2.238154e+31 2.572594e+29

3: predicted_arima 4.674561e+02 1.478031e+01 3.176420e-01

4: predicted_lr_arima 1.351231e+02 8.466822e+00 1.716606e-01

5: predicted_stlf 4.780604e+02 1.442692e+01 3.016014e-01

6: predicted_decom 5.249974e+02 1.810596e+01 4.927112e-01

Table 8: MSE, MAD, MAPE Values from Different models of Product 85004

From the table, the lowest MAPE value corresponds to ARIMA model. Thus, it’s the

best fit for the product.

Figure 16: Forecast in Units Over Time for Product 85004 with the Best Model Fit

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Figure 17: Forecast Comparison with Other Models for Product 85004

CONCLUSION

This report is aimed at determining the best models for predictions of daily sales for

eight different products which are currently being sold on one of the biggest e-commerce

companies in Turkey, Trendyol. By using the knowledge we gained from the Statistical

Forecasting and Time Series course, the necessary data analysis was performed for each

product, by converting their daily sales data to time series. For accurate forecasting, several

possible methods were being performed to find the best-fit candidate. These methods, which

are discussed in detail in the related chapter, were then compared with the train data to

determine accuracy. After determining the best model with fewer errors, the final forecasts

were made to predict the future sales of each product. To comment on results, it can be

achieved that any different data may have its own kind of features such as fluctuations, trends,

and as such. Different datasets may require different prediction models for accurate

forecasting, thus, each data should be treated distinctively when its future values is being

forecasted.

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APPENDIX

install.packages("jsonlite")

install.packages("httr")

install.packages("data.table")

install.packages("lubridate")

install.packages("forecast")

install.packages("ggplot2")

install.packages("xts")

install.packages("zoo")

# install.packages("tidyverse")

# install.packages("rlang")

# install.packages("dplyr")

#require(tidyverse)

require(jsonlite)

require(httr)

require(data.table)

require(forecast)

require(zoo)

require(xts)

require(lubridate)

require(ggplot2)

# require(dplyr)

# require(rlang)

# install.packages("tibble")

# require(tibble)

#sessionInfo()

library(xts)

library(timeDate)

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get_token <- function(username, password, url_site){

post_body = list(username,password)

post_url_string = paste0(url_site,'/token/')

result = POST(post_url_string, body = post_body)

# error handling (wrong credentials)

if(result$status_code==400){

print('Check your credentials')

return(0)

}

else if (result$status_code==201){

output = content(result)

token = output$key

}

return(token)

}

get_data<-function(start_date='2020-05-07', token, url_site){

post_body =

list(start_date=start_date,username="Group2",password="vahJADDus5zKAdHe")

post_url_string = paste0(url_site,'/dataset/')

header = add_headers(c(Authorization=paste('Token',token,sep=' ')))

result = GET(post_url_string, header, body = post_body)

output = content(result)

data = data.table::rbindlist(output)

data[,event_date:=as.Date(event_date)]

data = data[order(product_content_id,event_date)]

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return(data)

}

#get_token("Group2","vahJADDus5zKAdHe","http://167.172.183.67")

data<-get_data("2020-05-

07","a09e15e32a58726b702f78979d74dbda7c9594bf","http://167.172.183.67")

#Enter the product number to these two lines

product_chosen <- product_85004

chosen <- '85004'

product_chosen <-data[ which(data$product_content_id==chosen), ]

dates<-seq(as.Date("2019-04-30"),length=length(product_chosen$sold_count), by="days")

prod315_xts<-xts(product_chosen[,c("price","sold_count")], order.by = dates)

n.dates <- as.Date(c("2019-11-25", "2019-11-26", "2019-11-27","2019-11-28","2019-11-

29"))

black_friday <- as.xts(rep(1, 5), order.by = n.dates)

black_friday_1 <- merge(black_friday, prod315_xts, fill = 0)

prod315_xts <- merge(black_friday_1$black_friday, prod315_xts, fill = 0)

model_blackfriday <- lm(sold_count ~ price + black_friday, data =prod315_xts )

summary(model_blackfriday)

mean_315<-mean(product_chosen$sold_count[length(product_chosen)-

50:length(product_chosen)])

for(i in 1:length(prod315_xts$black_friday)) {

if(black_friday_1$black_friday[i]=="1"){

black_friday_1$sold_count[i]=mean_315

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i=i+1

}}

for(i in 1:length(prod315_xts$black_friday)) {

if(black_friday_1$black_friday[i]=="1"){

product_chosen$sold_count[i]=mean_315

i=i+1

}}

sold_count_1<-prod315_xts[,"sold_count"]

plot(sold_count_1)

black_friday_1$black_friday.1=NULL

train_315<-ts(black_friday_1$sold_count[1:(length(black_friday_1$sold_count)-20)],

frequency = 7)

test_315<-ts(black_friday_1$sold_count[(length(black_friday_1$sold_count)-

19):length(black_friday_1$sold_count)], frequency = 7)

stlf_model_315<-stlf(train_315,h=20)

stlf_model_315$mean

plot(stlf_model_315)

arima_model_315<-auto.arima(train_315,stepwise=FALSE)

summary(arima_model_315)

prediction_main=forecast_315<-forecast(arima_model_315,h=20)

plot(prediction_main)

test_data_315[,predicted_arima_main:=as.vector(prediction_main$mean)]

##....----------------------------------------------------------------------

decomposed_product_315_multiplicative<-decompose(train_315,type="multiplicative")

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plot(decomposed_product_315_multiplicative)

decomposed_product_315_additive<-decompose(train_315,type="additive")

plot(decomposed_product_315_additive)

decomposed_product_315_multiplicative$trend

first_trend_315=0

trend_model_315<- auto.arima(decomposed_product_315_multiplicative$trend,stepwise =

FALSE)

forecasted_trend_model_315<-forecast(trend_model_315, h=3)

plot(forecasted_trend_model_315)

decomposed_product_315_multiplicative$trend[1:3]=first_trend_315

decomposed_product_315_multiplicative$trend[(length(train_315)-

2):length(train_315)]=forecasted_trend_model_315$mean

plot(decomposed_product_315_multiplicative$trend)

model_product_315<- auto.arima(decomposed_product_315_multiplicative$trend, stepwise =

FALSE)

detrend_315=train_315/(decomposed_product_315_multiplicative$seasonal*decomposed_pr

oduct_315_multiplicative$trend)

detrend_315_model=auto.arima(detrend_315,stepwise = FALSE)

model_forecast_315=predict(detrend_315_model, n.ahead=20)$pred

model_forecast_315

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model_forecasted_315=model_forecast_315*(forecast(model_product_315,

h=20)$mean*c(decomposed_product_315_multiplicative$figure[3:7],decomposed_product_3

15_multiplicative$figure[1:7],decomposed_product_315_multiplicative$figure[1:7],decompo

sed_product_315_multiplicative$figure[1:1]))

model_forecasted_315

summary(model_forecasted_315)

plot(model_forecasted_315)

plot(product_chosen$sold_count, xlab ="Time",ylab = "Daily Sales", main="Number of

Product 31515569 Sales")

##.......

arima1<-auto.arima(train_315)

forecast_315<-forecast(arima1,h=20)

plot(forecast_315)

meanfor_315<-as.numeric(forecast_315$mean)

test_315_num<-as.numeric(test_315)

MAE<-mean(abs(meanfor_315-test_315_num))

MAPE<-100*mean(abs((meanfor_315 - test_315_num)/test_315_num))

print(MAPE)

for_dates<-seq(as.Date("2020-03-20"),length=20, by="days")

forecast315_xts<-xts(forecast_315$mean,order.by = for_dates)

plot(forecast315_xts)

plot(forecast315_xts, main = 'Forecast Comparison', ylim = c(0, 1500))

lines(forecast315_xts, col = "blue")

plot(test_315, main = 'Forecast Comparison', ylim = c(-100, 1600))

lower <- xts(forecast_315$lower[,2], order.by = for_dates)

upper <- xts(forecast_315$upper[,2], order.by = for_dates)

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lines(lower, col = "blue", lty = "dashed")

lines(upper, col = "red", lty = "dashed")

#.................................................

plot(product_chosen[,c("visit_count","sold_count")],main="Daily

Sales",xlab="Visit",ylab="Sales Quantity")

product_chosen[,day:=weekdays(event_date)]

product_chosen[,is_weekend:=ifelse(isWeekend(event_date),1,0)]

test_data_315=tail(product_chosen,20)

product_315=product_chosen[1:(nrow(product_chosen)-20)]#train yani

fit=lm(sold_count~visit_count,data=product_chosen)

fit

summary(fit)

checkresiduals(fit)

coef=fit$coefficients

coef

fit$residuals

plot(product_chosen[,c("visit_count","sold_count")],main="Daily

Sales",xlab="Visit",ylab="Sales Quantity")

abline(coef[1],coef[2],col="red")

plot(fit)

product_chosen[,logsales:=log(sold_count)]

product_chosen[,logvisit:=log(visit_count)]

is.na(product_chosen)<-sapply(product_chosen , is.infinite)

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fit=lm(logsales~logvisit+as.factor(is_weekend),data=product_chosen,na.action = na.omit)

length(is_weekend)

class(is_weekend)

summary(fit)

checkresiduals(fit)

test_data_315[,logsales:=log(sold_count)]

test_data_315[,logvisit:=log(visit_count)]

test_data_315[,logpredicted:=predict(fit,test_data_315)]

test_data_315[,predicted_slr:=exp(logpredicted)]

fit=lm(logsales~logvisit+as.factor(day),data=product_chosen,na.action = na.omit)

summary(fit)

checkresiduals(fit)

test_data_315[,logpredicted:=predict(fit,test_data_315)]

test_data_315[,predicted_lr:=exp(logpredicted)]

ts.plot(test_data_315$sold_count,ylim=c(50,900))

lines(test_data_315$predicted_slr,col="blue")

lines(test_data_315$predicted_lr,col="red")

### arima###

ts_315=ts(product_chosen[,c("sold_count")],frequency = 7)

ts.plot(ts_315,main="Daily Sales",xlab="Time",ylab="Sales Quantity")

arima_model=auto.arima(ts_315)

summary(arima_model)

checkresiduals(arima_model)

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model_fitted <- ts_315 - residuals(arima_model)

ts.plot(ts_315, xlab = "Time", ylab = "Daily Sales",main="Daily Sales")

points(model_fitted, type = "l", col = 2, lty = 2)

predictions=forecast(arima_model,h=20)

plot(predictions)

test_data_315[,predicted_arima:=as.vector(predictions$mean)]

ts.plot(test_data_315$sold_count,ylim=c(0,1000), xlab = "Time", ylab = "Daily

Sales",main="Daily Sales")

lines(test_data_315$predicted_slr,col="blue")

lines(test_data_315$predicted_lr,col="green")

lines(test_data_315$predicted_arima,col="red")

lines(as.numeric(stlf_model_315$mean), col="yellow")

lines(as.numeric(model_forecasted_315),col="pink")

lines(as.numeric(test_data_315$predicted_arima_main),col="brown")

test_data_315[,predicted_stlf:=as.numeric(stlf_model_315$mean)]

test_data_315[,predicted_decom:=as.numeric(model_forecasted_315)]

##dynamic##

data_ts=ts(product_chosen[,c("sold_count")],frequency = 7)

visits<-as.matrix(product_chosen[,"visit_count"])

arima_model_with_var <- auto.arima(data_ts,xreg= visits)

arima_model_with_var

plot(data_ts)

summary(arima_model_with_var)

checkresiduals(arima_model_with_var)

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model_fitted_2 <- data_ts - residuals(arima_model_with_var)

ts.plot(data_ts, xlab = "Time", ylab = "Daily Sales",main="Daily Sales")

points(model_fitted_2, type = "l", col = 2, lty = 2)

#predictions

visits_test<-as.matrix(test_data_315[,"visit_count"])

predictions=forecast(arima_model_with_var,h=20,xreg=visits_test)

plot(predictions)

test_data_315[,predicted_lr_arima:=as.vector(predictions$mean)]

ts.plot(test_data_315$sold_count,ylim=c(0,1000), xlab = "Time", ylab = "Daily

Sales",main="Daily Sales")

lines(test_data_315$predicted_slr,col="blue")

lines(test_data_315$predicted_lr,col="green")

lines(test_data_315$predicted_arima,col="red")

lines(as.numeric(stlf_model_315$mean), col="yellow")

lines(as.numeric(model_forecasted_315),col="purple")

lines(as.numeric(test_data_315$predicted_arima_main),col="brown")

lines(test_data_315$predicted_lr_arima,col="orange")

legend("topright", c("predicted_slr", "predicted_lr", "predicted_arima","stlf_model",

"predicted_lr_arima", "predicted_arima" ,"decomposed"), col =

c("blue","green","red","yellow","orange","brown" ,"purple"),

cex = 0.75, pch = 15)

pred_datatable=test_data_315[,.(event_date,sold_count,predicted_slr,predicted_lr,predicted_a

rima,predicted_lr_arima,test_data_315$predicted_arima_main,predicted_stlf,predicted_deco

m)]

melted=melt(pred_datatable,id.vars=c(1,2))

summary_result=melted[,list(se=(value-sold_count)^2,

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ad=abs(value-sold_count),

ape=abs(value-

sold_count)/sold_count,sold_count),by=list(event_date,variable)]

summary_result=summary_result[,list(mse=mean(se),mad=mean(ad),mape=mean(ape)),by=li

st(variable)]

summary_result

REFERENCES

1. www.datacamp.com